A generalization of surface family with common line of curvature

•We deduce the condition for the given curve to be the line of curvature on surface when the marching-scale functions are in more general expression.•Two functions θθ(s) and λλ(s) control the shape of the surface.•We classify the conditions by the expression of θθ(s).

We can represent the surface with a linear combination of the components of Frenet–Serret frame. Based on this representation, in the work of Li et al. [C.-Y. Li, R.-H. Wang, C.-G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des. 43 (9) (2011) 1110–1117], we derive the necessary and sufficient condition on the marching-scale functions for which the given curve is a line of curvature of the resulting surface. For convenience, we assumed the marching-scale functions can be decomposed into two factors. In this paper, we derive the sufficient condition for the given curve as a line of curvature of the surface when the marching-scale functions are in more general expressions. Finally, we give some representative examples to illustrate the convenience and efficiency of this method.